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A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration. (English) Zbl 0981.60078
Let \(\{B_t,t\geq 0\}\) be a one-dimensional standard Brownian motion starting from 0. Denote by \(B^{(\mu)}_t=B_t+\mu t\), \(t\geq 0\), a Brownian motion with constant drift \(\mu\) and consider the exponential functional \(A^{(\mu)}_t=\int _0^t \exp(2 B_s^{(\mu)}) ds\). The main task of this paper is the investigation of the relation – in terms of exponential – between \(B^{(-\mu)}\) and \(B^{(\mu)}\). For instance, the authors prove that the process \(\{1/A^{(-\mu)}_t, t>0\}\) has the same distribution as \(\{1/A^{(-\mu)}_t+ 1/\widetilde A^{(-\mu)}_{\infty}, t>0\}\) where \(\widetilde A^{(-\mu)}_{\infty}\) is a copy of \( A^{(-\mu)}_{\infty}\) independent of \(B^{(-\mu)}\). A number of variants and useful consequences of these results are obtained. The most important consequence expresses \(B^{(-\mu)}\) in terms \(B^{(\mu)}\) and an independent Gamma distribution. Two proofs of this result are presented. The first one is based on the theory of enlargements of the Brownian filtration and the other one is based on Lamperti’s relation and some properties of the laws of Bessel processes under time reversal and time inversion.

60J65 Brownian motion
60J55 Local time and additive functionals
60G15 Gaussian processes
60G48 Generalizations of martingales